OFFSET
0,1
COMMENTS
Also numbers of the form ((d*10^k)^2 + 2)/9^2 that are not squares, where d is a single-digit number.
The square roots of these numbers show runs of equal digits, see the link to Schizophrenic numbers.
LINKS
K. S. Brown, Schizophrenic numbers
Mathematics StackExchange, Is there any mathematical reason for this 'digit-repetition-show'?
Wikipedia, Schizophrenic Number
Index entries for linear recurrences with constant coefficients, signature (1,1000000000000000000,-1000000000000000000).
FORMULA
G.f.: (2/81)*(1/(1-x)+6249960/(1+1000000000*x)+6250040/(1-1000000000*x)).
a(n) = a(n-1) + 10^18*a(n-2) - 10^18*a(n-3).
a(2*n) = (25*10^(6 + 18*n) + 2)/81.
a(2*n + 1) = (16*10^(10 + 18*n) + 2)/81.
We use in the next formulas a special notation for real numbers where (x) after a digit denotes a run of length x for this digit. Example: 3(4).2(3) is 3333.2222 .
sqrt(a(2*n)) = 5(3+9*n).5(4+9*n)7(8+18*n)3(7+18*n)5(1)1(6+18*n)0(1)2(7+18*n)7(1)1(1)9(5+18*n)7(1)0(1)1(1)3(4+18*n)... .
sqrt(a(2*n+1)) = 4(5+9*n).4(6+9*n)7(1)2(10+18*n)1(1)3(1)5(1)4(1)1(1)6(6+18*n)7(1)2(1)0(1)9(1)2(1)0(1)1(1)3(1)8(4+18*n)... .
sqrt(1/a(2*n)) = 0.0(2+9*n)1(1)7(1)9(6+18*n)2(1)8(1)0(5+18*n)... .
sqrt(1/a(2*n+1)) = 0.0(4+9*n)2(2)4(1)9(8+18*n)8(1)5(1)9(1)3(1)7(1)5(1)0(5+18*n)... .
sqrt(a(2*n)-(2/81)) = 10^(4+9*n)/18.
sqrt(a(2*n+1)-(2/81)) = 10^(7+9*n)/225.
EXAMPLE
a(0) = 308642 and sqrt(a(0)) = 555.555577777777333333351111110222222271999997013333521... .
a(1) = 1975308642 and sqrt(a(1)) = 44444.4444447222222222213541666666720920138888465033637... .
PROG
(PARI) a(n) = (((5-(n%2))*10^(3+n*(9/2)-(n%2)*(5/2)))^2+2)/81
CROSSREFS
KEYWORD
nonn,easy,base
AUTHOR
Thomas Scheuerle, Jan 15 2023
STATUS
approved